Method and apparatus for creating a simulated particle pack

Abstract

A machine-implemented method is provided for placing a plurality of particles to create a simulated particle pack. The method comprises defining a central string, a space disposed about the central string, and N concentric subspaces disposed about the central string and within the space, each of the N subspaces corresponding to one of the N particle categories, selecting a particle from the plurality of particles, and placing the selected particle in the corresponding subspace so that the selected particle becomes a placed particle at a particle location unique to that placed particle and is in non-overlapping relation with other placed particles. The selected particle placement includes defining a catch net representative of buoyancy of a portion of the placed particles and positioning the catch net within the space based upon the placement of the portion of the placed particles. The selected particle placement further includes defining a water level representative of a level of a portion of the placed particles that are smaller than the selected particle and represent a surface of the smaller placed particles. The selected particle is placed in non-overlapping relation with respect to the catch net and the water level. The method further includes positioning the water level within the space based upon the smaller particle surface. The particle selection and placement procedures are repeated until a desired number of particles have been placed. Related apparatus also are disclosed.

1. A machine-implemented method for simulating placement of a plurality of unplaced particles, the method comprising:
 selecting a plurality of unplaced particles;  
 wherein each of the plurality of unplaced particles exhibits a characteristic dimension, corresponding to N categories of the plurality of unplaced particles;  
 wherein the characteristic dimension of each of the plurality of unplaced particles of a given category of the N categories is different from the characteristic dimension of each of the plurality of unplaced particles of other of the N categories;  
 defining a central string, a space disposed about the central string, and N subspaces disposed about the central string and within the space, wherein each of the N subspaces corresponds to one of the N categories, respectively;  
 selecting a particle from the plurality of unplaced particles;  
 defining a catch net representative of buoyancy of a portion of a plurality of placed particles and positioning the catch net within the space based upon placement of the portion of the plurality of placed particles;  
 defining a water level representative of a level of a portion of the plurality of placed particles that are smaller than the selected particle;  
 simulating placement of the selected particle within a corresponding subspace so that the selected particle is positioned in a non-overlapping relationship with respect to the plurality of placed particles, the catch net, and the water level; and  
 repeatedly selecting another particle from the plurality of unplaced particles and simulating placement thereof until placement of each of the plurality of unplaced particles has been simulated.  
 
     
     
       2. The method as recited in  claim 1 , wherein the N subspaces each comprise a cylindrical subspace positioned concentrically with respect to the central string. 
     
     
       3. A machine-implemented method for simulating placement of a plurality of unplaced particles, the method comprising:
 selecting a plurality of unplaced particles;  
 wherein each of the plurality of unplaced particles exhibits a characteristic dimension, corresponding to N categories of the plurality of unplaced particles;  
 wherein the characteristic dimension of each of the plurality of unplaced particles of a given category of the N categories is different from the characteristic dimension of each of the plurality of unplaced particles of other of the N categories;  
 defining a central string, a space disposed about the central string, and N subspaces disposed about the central string and within the space, wherein each of the N subspaces corresponds to one of the N categories, respectively;  
 selecting a particle from the plurality of unplaced particles;  
 defining a catch net representative of a buoyancy of a portion of a plurality of placed particles and positioning the catch net within the space based upon placement of the portion of the plurality of placed particles;  
 simulating placement of the selected particle within a corresponding subspace so that the selected particle is positioned in a non-overlapping relationship with respect to the plurality of placed particles and the catch net; and  
 repeatedly selecting another particle from the plurality of unplaced particles and simulating placement thereof until placement of each of the plurality of unplaced particles has been simulated.  
 
     
     
       4. The method as recited in  claim 3 , wherein each of the plurality of unplaced particles comprises a sphere and the characteristic dimension of each of the plurality of unplaced particles comprises a radius. 
     
     
       5. The method as recited in  claim 3 , wherein selecting the particle from the plurality of unplaced particles comprises randomly selecting the particle from the plurality of unplaced particles. 
     
     
       6. The method as recited in  claim 3 , further comprising:
 defining a pack surface for the plurality of placed particles; and  
 positioning the catch net relative to a position of the pack surface.  
 
     
     
       7. The method as recited in  claim 6 , wherein positioning the catch net relative to the pack surface comprises positioning the catch net at a distance away from the pack surface based upon a selected particle radius. 
     
     
       8. The method as recited in  claim 6 , wherein:
 the plurality of placed particles include a top layer; and  
 the position of the pack surface comprises an average of a particle location of each of the plurality of placed particles in the top layer thereof.  
 
     
     
       9. The method as recited in  claim 6 , wherein:
 each of the plurality of placed particles has a south pole;  
 the plurality of placed particles include a top layer; and  
 the pack surface corresponds to the south poles of the top layer of the plurality of placed particles.  
 
     
     
       10. The method as recited in  claim 9 , wherein the pack surface corresponds to an average of the south pole positions of each of the plurality of placed particles of the top layer. 
     
     
       11. The method as recited in  claim 6 , wherein the pack surface defining comprises:
 selecting a top layer of the placed particles;  
 for each particle category k of the placed particles in the top layer, defining a particle radius a i  for the placed particles i of that category k;  
 for subspace k corresponding to the particle category k, assigning a cylinder radius W k ;  
 assigning a top layer particle number m(k) and determining values for m(k) by evaluating 
                 ∑     i   =   1         m   ⁢     (   k   )       -   1           Submode   ⁢     (   i   )       ≤   k       ⁢     a   i   2       <     W   k   2     ≤         ∑     i   =   1       m   ⁢     (   k   )             Submode   ⁢     (   i   )       ≤   k       ⁢       a   i   2     ·   k         =   1     ,   2   ,   …   ⁢           ,   N       
 
 
       where N is a number of particle categories; and
 determining the pack surface location using 
       S   =       1   m     ⁢       ∑     i   =   1     m     ⁢     (       Z   i     -     a   i       )             
 
 
       where S represents the pack surface and Z i  represents a position of a center of a center one of the placed particles. 
     
     
       12. The method as recited in  claim 6 , wherein:
 for a given particle category k and corresponding subspace k, the particle placement comprises contacting an ith placed particle with the selected particle, the ith placed particle having a characteristic dimension a i  and the selected particle having a characteristic dimension a c ;  
 the catch net comprises a subnet corresponding to the subspace k; and  
 if a i /a c <1, then the catch net positioning comprises positioning the subnet k for a kth subspace Z net (k) at 
   Z net ( k )= S−a   i    
 
 
       where S represents the position of the pack surface. 
     
     
       13. The method as recited in  claim 6 , wherein:
 for a given particle category k and corresponding subspace k, the particle placement comprises contacting an ith placed particle with the selected particle, the ith placed particle having a characteristic dimension a i  and the selected particle having a characteristic dimension a c ;  
 the catch net comprises a subnet corresponding to the subspace k; and  
 if 1≦a i /a c <a x , where a x  represents a sample particle size for a corresponding sample particle that will fit into a cavity formed by placed spheres larger than the sample particle, then the catch net positioning comprises positioning a subnet k for a kth subspace Z net (k) at 
     Z   net ( k )= S −2 a   c    
 
 where S represents the position of the pack surface.  
 
     
     
       14. The method as recited in  claim 13 , wherein the sample particle size a x  is assigned a value of √6+2. 
     
     
       15. The method as recited in  claim 6 , wherein:
 for a given particle category k and corresponding subspace k, the particle placement comprises contacting an ith placed particle with the selected particle, the ith placed particle having a characteristic dimension a i  and the selected particle having a characteristic dimension a c ;  
 the catch net comprises a subnet corresponding to the subspace k; and  
 if a i /a c ≧a x , where a x  represents a sample particle size for a corresponding sample particle that will fit into a cavity formed by placed spheres larger than the sample particle, then the catch net positioning comprises positioning a subnet k for a kth subspace Z net (k) at 
     Z   net ( k )= S −2 a   c   −a   i    
 
 where S represents the position of the pack surface.  
 
     
     
       16. The method as recited in  claim 3 , wherein:
 the catch net extends across the space substantially perpendicularly to the central string.  
 
     
     
       17. The method as recited in  claim 3 , wherein:
 the catch net extends across each of the N subspaces substantially perpendicularly to the central string.  
 
     
     
       18. The method as recited in  claim 3 , wherein:
 each of the N subspaces extends substantially perpendicularly to the central string;  
 the catch net comprises N subnets; and  
 each of the N subnets corresponds to each of the N subspaces, respectively.  
 
     
     
       19. The method as recited in  claim 18 , wherein each of the N subnets extends across a corresponding subspace of the N subspaces. 
     
     
       20. The method as recited in  claim 19 , wherein:
 each of the N subnets has a level; and  
 the levels of at least two of the N subnets differ from one another.  
 
     
     
       21. The method as recited in  claim 20 , wherein positioning the catch net comprises positioning each of the N subnets at a selected distance from an end of the central string. 
     
     
       22. The method as recited in  claim 3 , wherein:
 the space includes a base surface; and  
 positioning the catch net comprises spacing the catch net away from the base surface.  
 
     
     
       23. The method as recited in  claim 22 , wherein spacing the catch net away from the base surface simulates a positioning of the catch net for a top layer of the placed particles. 
     
     
       24. The method as recited in  claim 3 , wherein positioning the catch net comprises:
 positioning the catch net for a k th  one of the N unplaced particle categories at a catch net position Z net (k) within a k th  one of the N subspaces determined by Z net (k)=Z init +H r a k  a min /a max ;  
 wherein:  
 Z init  is an initial catch net position for a k th  one of the N subspaces;  
 a k  is a characteristic dimension of the particles of a k th  one of the N unplaced particle categories;  
 a min  is a characteristic dimension of a small one of the particles;  
 a max  is a characteristic dimension of a large one of the particles;  
 r is a weighting coefficient; and  
 H is a switching coefficient.  
 
     
     
       25. The method as recited in  claim 24 , wherein the weighting coefficient is assigned a random number. 
     
     
       26. The method as recited in  claim 24 , wherein:
 below a threshold value the switching coefficient is assigned a value of one; and  
 above the threshold value the switching coefficient is assigned a value of zero.  
 
     
     
       27. An apparatus for simulating placement of a plurality of unplaced particles, comprising:
 an input device for inputting particle selection information;  
 a storage device operatively coupled to the input device for storing the particle selection information; and  
 a processor for repeatedly selecting a particle from the plurality of unplaced particles, for simulating placement of the selected particle within a space in a non-overlapping relationship with respect to previously placed particles to form a plurality of placed particles, and for establishing a catch net representative of buoyancy of a portion of the plurality of placed particles and positioning the catch net within a space based upon placement of the portion of the plurality of placed particles, the processor configured for placing the selected particle in non-overlapping relationship with respect to the catch net.  
 
     
     
       28. A machine-readable medium for use in simulating the placement of a plurality of unplaced particles, comprising:
 machine executable instructions for defining a central string, a space disposed about the central string, and N subspaces disposed about the central string and within the space, each of the N subspaces corresponding to one of N categories of the plurality of particles, wherein each of the N categories corresponds to a characteristic dimension of the plurality of unplaced particles;  
 wherein the machine executable instructions are configured for repeatedly selecting a particle from the plurality of unplaced particles and simulating placement of the selected particle in a subspace of the N subspaces corresponding to the one category of N categories in a non-overlapping relationship with previously placed particles, to form a plurality of placed particles; and  
 wherein the machine executable instructions are configured for defining a catch net representative of buoyancy of a portion of the plurality of placed particles and positioning the catch net within the space based upon the placement of the portion of the plurality of placed particles, the selected particle being placed in non-overlapping relation with respect to the catch net.  
 
     
     
       29. A machine-implemented method for simulating placement of a plurality of particles, the method comprising:
 selecting a plurality of unplaced particles;  
 wherein each of the plurality of unplaced particles exhibits a characteristic dimension, corresponding to N categories of the plurality of unplaced particles;  
 wherein the characteristic dimension of each of the plurality of unplaced particles of a given category of the N categories is different from the characteristic dimension of each of the plurality of particles of other of the N categories;  
 defining a central string, a space disposed about the central string, and N subspaces disposed about the central string and within the space, wherein each of the N subspaces corresponds to one of the N categories;  
 selecting a particle from the plurality of unplaced particles;  
 defining a water level representative of a level of a portion of a plurality of placed particles that are smaller than the selected particle;  
 simulating placement of the selected particle within a corresponding subspace so that the selected particle is positioned in a non-overlapping relationship with respect to the plurality of placed particles and the water level; and  
 repeatedly selecting another particle from the plurality of unplaced particles and simulating placement thereof until each of the particles of the plurality of unplaced particles has been simulated.  
 
     
     
       30. The method as recited in  claim 29 , wherein each of the plurality of unplaced particles comprises a sphere and the characteristic dimension of each of the plurality of unplaced particles comprises a radius. 
     
     
       31. The method as recited in  claim 29 , wherein:
 defining the water level comprises determining an average location of the particle locations along the central string of the plurality of placed particles and positioning the water level at the average location.  
 
     
     
       32. The method as recited in  claim 29 , wherein:
 the water level comprises a plurality of subspace water levels wherein each of the plurality of subspace water levels corresponds to one of the N subspaces, respectively; and  
 further comprising positioning the water level comprising positioning each of the plurality of subspace water levels.  
 
     
     
       33. The method as recited in  claim 32 , wherein:
 each of the N subspaces comprises a subspace surface representative of a portion of placed particles therein, each of the subspace surfaces comprising a subspace surface location with respect to the central string;  
 each of the plurality of placed particles comprises a north pole having a north pole location; and  
 the subspace water level position for one of the subspaces is determined by determining an average location of the north pole locations of the portion of the plurality of placed particles within the one subspace and assigning the average location of the subspace surface location for the one subspace.  
 
     
     
       34. The method as recited in  claim 32 , wherein:
 each of the N subspaces comprises a subspace surface representative of a portion of smaller placed particles within that subspace, each of the subspace surfaces comprising a subspace surface location with respect to the central string;  
 each of the plurality of placed particles comprises a south pole having a south pole location; and  
 the subspace water level position for one of the subspaces is determined by determining an average location of the south pole locations of the portion of the plurality of placed particles within the one subspace and assigning the average location as the subspace surface location for the one subspace.  
 
     
     
       35. The method as recited in  claim 32 , wherein:
 positioning the water level comprises using an offset for each of the subspace water level positions.  
 
     
     
       36. The method as recited in  claim 29 , wherein:
 defining the water level comprises using an offset to position the water level.  
 
     
     
       37. An apparatus for simulating placement of a plurality of unplaced particles, comprising:
 an input device for inputting particle selection information;  
 a storage device operatively coupled to the input device for storing the particle selection information; and  
 a processor for selecting a particle from the plurality of particles, for placing the selected particle in the corresponding subspace so that the selected particle becomes a placed particle at a particle location unique to that placed particle and is in non-overlapping relation with other placed particles, for establishing a water level representative of a level of a portion of the placed particles that are smaller than the selected particle and represent a surface of the smaller placed particles, and for positioning the water level within the space based upon the smaller particle surface, the selected particle being placed in non-overlapping relation with respect to the water level.  
 
     
     
       38. A machine-readable medium for use in simulating placement of a plurality of unplaced particles, comprising:
 machine executable instructions for defining a central string, a space disposed about the central string, and N subspaces disposed about the central string and within the space, each of the N subspaces corresponding to one of N particle categories;  
 wherein the machine executable instructions are configured for repeatedly selecting a particle from the plurality of particles and simulating the placement of the selected particle in a subspace of the N subspaces of the corresponding category of N categories in non-overlapping relation with previously placed particles, to form a plurality of placed particles; and  
 wherein the machine executable are configured for defining a water level representative of a portion of the plurality of placed particles that are smaller than the selected particle and represent a surface of the portion of the plurality of placed particles and for positioning the water level within the space based upon the smaller particle surface, the selected particle being placed in non-overlapping relation with respect to the water level.  
 
     
     
       39. A machine-implemented method for simulating the placement of a particle, the method comprising:
 defining a central string and a space having a cylindrical boundary wall disposed about the central string;  
 defining a plurality of cylindrical subspaces disposed about the central string and within the space, wherein each of the subspaces has a cylindrical boundary wall;  
 wherein the space includes a plurality of previously placed particles;  
 defining a water level representative of a level of a portion of the plurality of previously placed particles having a size which is smaller than a selected particle;  
 simulating movement of a particle within the space from an initial position and in a selected direction;  
 simulating contact of the particle with at least one of the cylindrical boundary wall of the space, the water level, the cylindrical wall of one of the plurality of subspaces, and at least one of the plurality of previously placed particles; and  
 determining stable placement of the particle within the space.  
 
     
     
       40. The method as recited in  claim 39 , wherein simulating contact of the particle with at least one of the cylindrical boundary wall of the space, the water level, the cylindrical boundary wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles comprises simulating rolling of the particle with respect to at least one of the cylindrical boundary wall of the space, the water level, the cylindrical wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles. 
     
     
       41. The method as recited in  claim 39 , wherein simulating contact of the particle with at least one of the cylindrical boundary wall of the space, the water level, the cylindrical boundary wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles comprises determining whether contact of the particle with at least one of the cylindrical boundary wall of the space, the water level, the cylindrical boundary wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles is compressive or tensile. 
     
     
       42. The method as recited in  claim 39 , wherein simulating movement of the particle within the space from the initial position and in the selected direction comprises simulating movement of the particle within the space in a direction parallel to the central string. 
     
     
       43. The method as recited in  claim 39 , wherein simulating movement of the particle within the space from the initial position and in the selected direction comprises constraining a center of the particle to remain within one of the plurality of cylindrical subspaces during simulating movement thereof. 
     
     
       44. The method as recited in  claim 39 , further comprising:
 defining a catch net representative of buoyancy of a portion of the plurality of placed particles and positioning the catch net within the space based upon the placement of the portion of the plurality of placed particles.  
 
     
     
       45. A machine-implemented method for simulating the placement of a particle, the method comprising:
 defining a central string and a space having a cylindrical boundary wall disposed about the central string;  
 defining a plurality of cylindrical subspaces disposed about the central string and within the space, wherein each of the subspaces has a cylindrical boundary wall;  
 wherein the space includes a plurality of previously placed particles;  
 defining a catch net representative of a buoyancy of a portion of the plurality of previously placed particles and positioning the catch net within the space based upon the placement of the portion of the plurality of previously placed particles;  
 simulating movement of a particle within the space from an initial position and in a selected direction;  
 simulating contact of the particle with at least one of the cylindrical boundary wall of the space, the catch net, the cylindrical boundary wall of one of the plurality of subspaces, and at least one of the plurality of previously placed particles; and  
 determining stable placement of the particle within the space.  
 
     
     
       46. The method as recited in  claim 45 , wherein simulating contact of the particle with at least one of the cylindrical boundary wall of the space, the catch net, the cylindrical boundary wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles comprises simulating rolling of the particle with respect to at least one of the cylindrical boundary wall of the space, the catch net, the cylindrical boundary wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles. 
     
     
       47. The method as recited in  claim 45 , wherein simulating contact of the particle with at least one of the cylindrical boundary wall of the space, the catch net, the cylindrical boundary wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles comprises determining whether contact of the particle with at least one of the cylindrical boundary wall of the space, the catch net, the cylindrical boundary wall of one of the plurality of subspaces, and the at least one of the plurality of previously placed particles is compressive or tensile. 
     
     
       48. The method as recited in  claim 45 , wherein simulating movement of the particle within the space from the initial position and in the selected direction comprises simulating movement of the particle within the space in a direction parallel to the central string. 
     
     
       49. The method as recited in  claim 45 , wherein simulating movement of the particle within the space from the initial position and in the selected direction comprises constraining a center of the particle to remain within one of the plurality of cylindrical subspaces during simulating movement thereof. 
     
     
       50. The method as recited in  claim 45 , further comprising:
 defining a water level representative of a level of a portion of the plurality of previously placed particles having a size which is smaller than the selected particle.